ON TEMPORAL INSTABILITY

OF ELECTRICALLY FORCED JETS WITH NONZERO

BASIC STATE VELOCITY

Sayantan Das(SD)

Masters Student @ UT Pan Am

Mentors :Dr . D.N. Riahi & Dr. D. Bhatta

IN OTHER WORDS…

Modeling instabilities of the Electro spinning process

WHAT IS ELECTRO -SPINNING?

Process of producing nano-fibers

http://nano.mtu.edu/Elect

rospinning_start.html

QUALITY NANOFIBERS

unparalleled in their porosity, high surface area

fineness and uniformity

STABILITY?

. Here stability in terms of perturbation is considered

IN DETAIL

Schematic Representation

To detect and understand temporal

instabilities

Parameter regime under which

Instabilities are strong

Subsequent ways to control and eliminate such instabilities

WHY

WE USE,ELECTRO-HYDRODYNAMIC

EQUATION mass conservation D/Dt+.u=0

(1a) momentum Du/Dt =P+.( u)+qE

(1b) charge conservation Dq/Dt+.(KE)=0

(1c) electric potential E=

(1d) D/Dt =/t+ u. - total derivative t-time variable

u-velocity vector P-pressure E -

electric field vector -electric potential q- charge

-fluid density -dynamic viscosity K-conductivity

HOW WE MODEL?

We non- dimensionaliz

e 1(a-d)

We get four non

dimensional equation 2(a-

d)

Using perturbation technique we linearize the

PDE,s

Forming 4x4

determinant , we get the Dispersi

on relation

THE NON DIMENSIONAL EQN

022

vhz

ht

(2a)

2

2

3

2

2

2

24

811

E

z

h

z

h

z

h

h

zz

vv

t

v

z

vh

zhh

E 22

32

(2c)

hz

Ehz

EzEb 42

)ln()( 22

2

(2d)

 

02

1 2

KEhz

hvz

ht

(2b)

  All the constant

parameters are from

Hohman et al 2001

PERTURBATION TECHNIQUE

We consider (h,v, , E)=

Perturbation quantities , by subscript ‘1’

Where, =()exp(

)are assumed to be small in magnitude

Basic state solution , by subscript ‘b’

Linearized w.r.t. amplitude

The complex growth rate ()

k is the axial wave number

MATHEMATICALLY

We plug in (h,v, , E) in the non dimensional equation

We then get the coefficient of each dependent

variable for each equations

Then we form a 4X4 Determinant of the coefficients .

Then by finding a nontrivial solution , we find the

DISPERSION RELATION

DISPERSION RELATION tells us about the growth

rate &frequency of the perturbations

OUR WORK

Hohman et.al ,2001, considered the basic state

velocity to be zero

We considered basic state velocity to be a non

zero and a constant quantity

Considering this case we derived the

DISPERSION RELATION

DISPERSION RELATION

Where,

0322

13 TTT

K

ivkkT b

431

41

44

12

2

1 22

22

2

b

b

EKkkT

k

Kkiviv bb

863

212

442

14 2222

3

k

EiEkKkT bb

bb

KEk

vik bb

b

121

44

42

1 222

3

bb ikvK

kvk

43 222

2,),(ln,89.0

1 2 kk

We get ,

with;

COMPUTATIONAL

We use Matlab to produce the zeroes of the

dispersion relation

In Matlab we used the inbuilt function Fzero

Fzero finds the root of a function

For growth rate we considered the real part of

For frequency we considered the imaginary part

of

RESULTS

Growth rate v/s Wave number for K*=inf ,vb=1, and variable applied field

Growth rate v/s Wave number for K*=0,vb=1,and variable applied field

MORE…

Growth rate v/s Wave number for K*=19.3 ,vb=1, for variable applied field

Growth rate v/s Wave number for K*=19.3,v*=0.3,sigmab=0.1vb=1,Eb varied

Contd….

Primary and Secondary modes with K*=19.3,sigmab=0.1,vb=1,Eb=2.9,&v*=0

SO…

• The variable applied field is stabilizing

• The finite values of either viscosity or conductivity are stabilizing

• There are two modes of instability for small values of the wavenumber

• All above results comply with Hohman et al with zero basic state velocity

• Hence,the growth rate in temporal instabilty is unaffected by the value of the basic

state velocity, but significant changes are already seen in spatial instability cases.

•So is our work is of no importance ? with vb being nonzero

NO

The non zero basic state velocity significantly

affects the frequency of the perturbed state

Hence also affects the period

Which is significant for producing quality fibers

LETS SEE HOW

FIGURE

Frequency v/s k, with K*=0,v*=0,sigmab=0.1,Eb=2.9

Frequency v/s k, with K*=19.3,v*=0,sigmab=0.1,Eb=2.9

HENCE

More the vb less is the frequency , hence more is

the period

Presence of conductivity increases the period

As velocity of the wave is proportional to the

negative frequency

As vb increases the velocity of the wave increases

(Obvious)

Hence production of nanofibers will be affected

FUTURE STUDIES…

• Investigate the case for spatial instability with

non zero basic state velocity

• Investigate combined spatial and temporal

instability with non zero basic state velocity

• Investigate non-linear model

• Investigate non axisymmetric case

THANK YOU ALL…

My special thanks to Dr Bhatta, & Dr Riahi for the support

and enthusiasm…..

Any questions or comments are gladly welcomed